cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A327934 Numbers k such that there is no prime p such that p^p divides k, but for its arithmetic derivative a positive finite number of such primes exist.

Original entry on oeis.org

15, 35, 39, 51, 55, 87, 91, 95, 111, 115, 119, 123, 143, 155, 158, 159, 183, 187, 203, 215, 219, 225, 235, 247, 259, 267, 275, 287, 291, 295, 299, 303, 319, 323, 327, 329, 335, 339, 355, 371, 374, 391, 395, 403, 407, 410, 411, 415, 427, 441, 447, 451, 471, 473, 482, 511, 515, 519, 525, 527, 533, 535, 543, 551, 559, 579
Offset: 1

Views

Author

Antti Karttunen, Oct 01 2019

Keywords

Comments

a(15) = 158, being even, is the first term not in A080774 or A202237, while 275 (= 5^2 * 11) is the first odd term not in the latter.
After 1, 3375 (= 3^3 * 5^3) is the first term in A202237 that is not present in this sequence, even though the overwhelming majority of the terms of A202237 are also included here. On the other hand, A080774 is a genuine subsequence of this sequence, as the sum of prime factors of such semiprimes is always a multiple of 4.
In contrast to A360110, this is not a multiplicative semigroup; For example, although 15 = 3*5 and 275 = 5^2 * 11 are both present, their product 15*275 = 4125 = 3 * 5^3 * 11 is not. - Antti Karttunen, Jan 31 2023

Examples

			From _Antti Karttunen_, Feb 01 2023: (Start)
1 has arithmetic derivative 1' = A003415(1) = 0. There are an infinite number of primes p such that p^p | 0, but because infinity is not a finite number, 1 is excluded from this sequence.
275 = 5^2 * 11 has no divisor of the form p^p, but its arithmetic derivative 275' = 135 = 3^3 * 5 has one divisor of the form p^p, therefore 275 is included in this sequence.
4125 = 3 * 5^3 * 11 has arithmetic derivative 4125' = A003415(4125) = 4225 = 5^2 * 13^2, that has no divisor of the form p^p, therefore 4125 is not included in this sequence.
(End)

Links

Crossrefs

Intersection of A048103 and A327929.
Cf. A003415, A099309, A129251, A202237, A256750, A327932, A359550, A360110, A360111 (characteristic function).
Cf. A080774 (subsequence).

Programs

Extensions

Definition clarified to exclude the term 1 by Antti Karttunen, Feb 01 2023

A072202 Same numbers of prime factors of forms 4*k+1 and 4*k+3, counted with multiplicity.

Original entry on oeis.org

1, 2, 4, 8, 15, 16, 30, 32, 35, 39, 51, 55, 60, 64, 70, 78, 87, 91, 95, 102, 110, 111, 115, 119, 120, 123, 128, 140, 143, 155, 156, 159, 174, 182, 183, 187, 190, 203, 204, 215, 219, 220, 222, 225, 230, 235, 238, 240, 246, 247, 256, 259, 267, 280, 286, 287, 291
Offset: 1

Views

Author

Reinhard Zumkeller, Jul 03 2002

Keywords

Comments

Equivalently, numbers n such that A083025(n) = A065339(n), indices of zeros in A079635.
Closed under multiplication.
Closed with respect to permutation A267099. - Antti Karttunen, Feb 03 2016

Examples

			825 = 3*5*5*11 = [(4*0+3)*(4*2+3)]*[(4*1+1)*(4*1+1)], therefore 825 is a term.

Links

Crossrefs

Cf. A002144, A002145, A202237 (odd elements), A079635, A065339, A083025, A267099.
Primitive elements are {2} U A080774. - Franklin T. Adams-Watters, Dec 16 2011.
Subsequence of A078613 and of A268381.

Programs

  • Haskell
    a072202 n = a072202_list !! (n-1)
a072202_list = [x | x <- [1..], a083025 x == a065339 x]
-- Reinhard Zumkeller, Jan 10 2012
  • Mathematica
    f[n_]:=Plus@@Last/@Select[If[==1,{},FactorInteger[n]],Mod[#[[1]],4]==1&]; Table[f[n],{n,100}] (* Ray Chandler, Dec 18 2011 *)
  • PARI
    isok(n) = {my(f = factor(n)); sum(k=1, #f~, ((f[k,1] % 4)==1)*f[k,2]) == sum(k=1, #f~, ((f[k,1] % 4)==3)*f[k,2]);} \\ Michel Marcus, Feb 05 2016
  • Scheme
    (define A072202 (ZERO-POS 1 1 A079635)) ;; [requires also my IntSeq-library] - Antti Karttunen, Feb 03 2016
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